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Some New Hilbert's Type Inequalities
Journal of Inequalities and Applications volume 2009, Article number: 851360 (2009)
Abstract
Some new inequalities similar to Hilbert's type inequality involving series of nonnegative terms are established.
1. Introduction
In recent years, several authors [1–10] have given considerable attention to Hilbert's type inequalities and their various generalizations. In particular, in [1], Pachpatte proved somenew inequalities similar to Hilbert's inequality [11, page 226] involving series of nonnegative terms. The main purpose of this paper is to establish their general forms.
2. Main Results
In [1], Pachpatte established the following inequality involving series of nonnegative terms.
Theorem 2 A.
Let and let and be two nonnegative sequences of real numbers defined for and , where are natural numbers. Let and . Then
where
We first establish the following general form of inequality (2.1).
Theorem 2.1.
Let and . Let and be positive sequences of real numbers defined for and , where are natural numbers. Let and Then
where
Proof.
By using the following inequality (see [12]):
where is a constant, and , , we obtain
Similarly, we have
From (2.6) and (2.7), using Hölder's inequality [13] and the elementary inequality:
where , and we have
Dividing both sides of (2.9) by summing up over from 1 to first, then summing up over from 1 to , using again Hölder's inequality, then interchanging the order of summation, we obtain
This completes the proof.
Remark 2.2.
Taking , (2.3) becomes
Taking and changing , , and into , , and respectively, and with suitable changes, (2.11) reduces to Pachpatte [1, inequality (1)].
In [1], Pachpatte also established the following inequality involving series of nonnegative terms.
Theorem 2 B.
Let , be as defined in Theorem A. Let and be positive sequences for and where are natural numbers. Define and . Let and be real-valued, nonnegative, convex, submultiplicative functions defined on Then
where
Inequality (2.12) can also be generalized to the following general form.
Theorem 2.3.
Let , , and be as defined in Theorem 2.1. Let and be positive sequences for and . Define and Let and be real-valued, nonnegative, convex, submultiplicative functions defined on Then
where
Proof.
By the hypotheses, Jensen's inequality, and Hölder's inequality, we obtain
Similarly,
By (2.16) and (2.17), and using the elementary inequality:
where and we have
Dividing both sides of (2.19) by and summing up over from 1 to first, then summing up over from 1 to , using again inverse Hölder's inequality, and then interchanging the order of summation, we obtain
The proof is complete.
Remark 2.4.
Taking , (2.14) becomes
where
Taking and changing , , and into , , and respectively, and with suitable changes, (2.21) reduces to Pachpatte [1, Inequality ( 7)].
Theorem 2.5.
Let , , , , and , be as defined in Theorem 2.3. Define
for and where are natural numbers. Let and be real-valued, nonnegative, convex functions defined on Then
Proof.
By the hypotheses, Jensen's inequality, and Hölder's inequality, it is easy to observe that
Proceeding now much as in the proof of Theorems 2.1 and 2.3, and with suitable modifications, it is not hard to arrive at the desired inequality. The details are omitted here.
Remark 2.6.
In the special case where and , Theorem 2.5 reduces to the following result.
Theorem 2 C.
Let be as defined in Theorem B. Define and for and , where are natural numbers. Let and be real-valued, nonnegative, convex functions defined on Then
This is the new inequality of Pachpatte in [1,Theorem 4].
Remark 2.7.
Taking and in Theorem 2.5, and in view of , we obtain the following theorem.
Theorem 2 D.
Let , be as defined in Theorem A. Define and for and , where are natural numbers. Let and be real-valued, nonnegative, convex functions defined on Then
This is the new inequality of Pachpatte in [1,Theorem 3].
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Acknowledgments
Research is supported by Zhejiang Provincial Natural Science Foundation of China(Y605065), Foundation of the Education Department of Zhejiang Province of China (20050392). Research is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P) and a HKU Seed Grant forBasic Research.
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Zhao, CJ., Cheung, WS. Some New Hilbert's Type Inequalities. J Inequal Appl 2009, 851360 (2009). https://doi.org/10.1155/2009/851360
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DOI: https://doi.org/10.1155/2009/851360